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प्रश्न
If `f(x) = (5^x + 5^-x - 2)/(x^2)` for x ≠ 0
= k for x = 0
is continuous at x = 0, find k
उत्तर
Function f is continuous at x = 0
∴ `"f"(0) = lim_(x→0) "f"(x)`
∴ k = `lim_(x→0) (5^x + 5^-x - 2)/(x^2)`
= `lim_(x→0) (5^x + 1/5^x - 2)/(x^2)`
= `lim_(x→0) ((5^x)^2 + 1 - 2(5^x))/(5^x*x^2)`
= `lim_(x→0) ((5^x - 1)^2)/(5^x*x^2)` ...[∵ a2 - 2ab + b2 = (a - b)2]
= `lim_(x→0) ((5^x - 1)/x)^2. 1/5^x`
= `lim_(x→0) ((5^x - 1)/x)^2 xx lim_(x→0) 1/5^x`
= `(log 5)^2 xx 1/5^0 ....[∵ lim_(x→0)(("a"^x - 1)/x) = log "a"]`
∴ k = (log 5)2
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